Seeing what we have with the show I think I have it more or less worked out. I'm thinking the Others will be defeated first and the story will end cleaning up who will rule what, but it's not going to be a 'the Whitewalkers were a red herring, the real story was always about the Throne' thing that you're getting from the TV thread.
A key point of distinction between the two is the Night King. I think the Others being a creation of the Children of the Forest will stick, but with the show creating the Night King they created a single point of emphasis on the threat where they could drop all the history and lore and defeat it with an assassins Valyrian steel dagger. The Night King doesn't exist in the same way in the books, the "Others" were beings that were sealed in a tomb who got let out because Mance Rayder went digging for treasure - this isn't in the show. It's also been implied that the Others are directly related to the wonky winter weather schedule.
With the Others rising to strength Westeros is due for another "Long Night". The winter in the books is going to be far more dreadful than the occasional slight dusting they showed in the show, it will threaten everyone and the only way out is to tie together all the prophecies and build up and end the threat once and for all. This is basically the "Winds of Winter" which seems like it should be two books, especially since you got a number of chapters that couldn't be fit into Dance dumped into it (Dance+Feast+these chapters maybe should have been 3 books).
Which finally leads us to A Dream of Spring. People have been going back and forth on the influence The Scouring of the Shire has on the series, and I think it would similar to that but not really the same thing. Rather this is the Song of Ice and Fire, Jon and Dany's story that will eventually intertwine, and Dream isn't about who sits the Iron Throne but rather the conclusion of this 'song'. That you don't just get a happy ending after defeating the big bad guys, rather it's how Jon and Dany cope in the aftermath leading to a "bittersweet" ending - like Dany goes nuts and burns everything so Jon has to kill her